Prethermalization

Heavy ion collision Heavy ion collision

Hadron abundancies Hadron abundancies RHIC

follow thermal eqilibrium distribution for suitable temperature T and baryon chemical potential μ

Hadron abundancies in e + - e - collisions Becattini

not thermal equilibrium ! no substantial scattering of produced hadrons no Boltzmann equations apply

Prethermalization J.Berges, Sz.Borsanyi, CW

Thermal equilibrium only one parameter characterizes distribution functions and correlations : temperature T ( several parameters if other quantities besides energy E are conserved, e.g. chemical potential μ or particle density n =N/V for conserved particle number N ) ( several parameters if other quantities besides energy E are conserved, e.g. chemical potential μ or particle density n =N/V for conserved particle number N )

essence of thermalization loss of memory of details of initial state for fixed volume V : only energy E matters

prethermalization only partial loss of memory of initial state

prethermalization can happen on time scales much shorter than for thermalization can happen on time scales much shorter than for thermalization nevertheless: several important features look already similar to thermal equilibrium state nevertheless: several important features look already similar to thermal equilibrium state can produce states different from thermal equilibrium that persist for very long ( sometimes infinite ) time scales can produce states different from thermal equilibrium that persist for very long ( sometimes infinite ) time scales

quantities to investigate set of correlation functions set of correlation functions or effective action as generating functional for correlation functions or effective action as generating functional for correlation functions not: density matrix or probability distribution not: density matrix or probability distribution

reasons rather different density matrices / probability distributions can produce essentially identical correlation functions rather different density matrices / probability distributions can produce essentially identical correlation functions ( they only differ by unobservable higher order correlations e.g point functions, or unobservable phase correlations ) ( they only differ by unobservable higher order correlations e.g point functions, or unobservable phase correlations ) only correlation functions are observed in practice only correlation functions are observed in practice ( distributions : one-point function or expectation value ) ( distributions : one-point function or expectation value )

Boltzmann’s conjecture start with arbitrary initial probability distribution start with arbitrary initial probability distribution wait long enough wait long enough probability distribution comes arbitrarily close to thermal equilibrium distribution probability distribution comes arbitrarily close to thermal equilibrium distribution probably not true ! probably not true ! but : observable correlations come arbitrarily close to thermal equilibrium values ( not all systems ) but : observable correlations come arbitrarily close to thermal equilibrium values ( not all systems )

time flow of correlation functions prethermalized state = partial fixed point in space of correlation functions

time evolution of correlation functions

hierarchical system of flow equations for correlation functions BBGKY- hierarchy Yvon, Born, Green, Kirkwood, Bogoliubov for interacting theories : system not closed

non-equilibrium effective action variation of effective action yields field equations in presence of fluctuations, and time dependent equal-time correlation functions

exact evolution equation

cosmology : evolution of density fluctuations J.Jaeckel,O.Philipsen,…

baryonic accoustic peaks Pietroni, Matarrese

classical scalar field theory ( d=1) G.Aarts,G.F.Bonini,… can be solved numerically by discretization on space-lattice

thermalization of correlation functions modetemperature G.Aarts, G.F.Bonini, CW, 2000

classical scalar field theory ( d=1) G.Aarts,G.F.Bonini,…

truncation momentum space

partial fixed points further truncation : momentum independent u,v,w,y,z (N-component scalar field theory, QFT)

comparison of approximations

quantum field theory

extensions (1) non-equal time correlation functions (2) 2PI instead 1PI ( J.Berges,…) permits contact to Schwinger-Keldish formalism

Hadron abundancies in heavy ion collisions Hadron abundancies in heavy ion collisions RHIC

Is temperature defined ? Does comparison with equilibrium critical temperature make sense ?

Prethermalization J.Berges,Sz.Borsanyi,CW

Vastly different time scales for “thermalization” of different quantities here : scalar with mass m coupled to fermions ( linear quark-meson-model ) ( linear quark-meson-model ) method : two particle irreducible non- equilibrium effective action ( J.Berges et al )

Thermal equilibration : occupation numbers Thermal equilibration : occupation numbers

Prethermalization equation of state p/ε Prethermalization equation of state p/ε similar for kinetic temperature

different “temperatures” different “temperatures”

Mode temperature n p :occupation number for momentum p late time: Bose-Einstein or Fermi-Dirac distribution

Kinetic equilibration before chemical equilibration Kinetic equilibration before chemical equilibration

Once a temperature becomes stationary it takes the value of the equilibrium temperature. Once chemical equilibration has been reached the chemical temperature equals the kinetic temperature and can be associated with the overall equilibrium temperature. Comparison of chemical freeze out temperature with critical temperature of phase transition makes sense

isotropization two-point functions for momenta in different directions

isotropization occurs before thermalization occurs before thermalization different time scale different time scale gradient expansion, Boltzmann equations become valid only after isotropization gradient expansion, Boltzmann equations become valid only after isotropization

some questions

Can pre-thermalized state be qualitatively different from thermal equilibrium state ? yes yes e.g. e + - e - collisions : e.g. e + - e - collisions : particle abundancies close to thermal, momentum disributions not

Is there always a common temperature T in pre-thermalized state ? no no e.g. two components with weak coupling

Does one always reach thermal equilibrium for time going to infinity ? no no simple obstructions : initial energy distribution exact non-thermal fixed points are possible instabilities from long-range forces ( gravity ) in practice : metastable states

role and limitation of linear response theory ? fails for approach to thermal equilibrium fails for approach to thermal equilibrium time scales of linear response are often characteristic scales for prethermalization time scales of linear response are often characteristic scales for prethermalization

conclusions Approach to thermal equilibrium is a complex process involving very different time scales. Approach to thermal equilibrium is a complex process involving very different time scales. This holds already for simple models as scalar field theory. This holds already for simple models as scalar field theory. Observation sees often only early stages, not equilibrium state : prethermalization. Observation sees often only early stages, not equilibrium state : prethermalization. Prethermalization can be characterized by partial fixed points in flow of correlation functons. Prethermalization can be characterized by partial fixed points in flow of correlation functons.

time flow of correlation functions prethermalized state = partial fixed point in space of correlation functions

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